Resonantly forced gravity-capillary lumps on deep water. Part 1. Experiments

The wave pattern generated by a pressure source moving over the free surface of deep water at speeds, U, below the minimum phase speed for linear gravity-capillary waves, c(min), was investigated experimentally using a combination of photographic measurement techniques. In similar experiments, using a single pressure amplitude, Diorio et al. (Phys. Rev. Lett., vol. 103, 2009, 214502) pointed out that the resulting surface response pattern exhibits remarkable nonlinear features as U approaches c(min), and three distinct response states were identified. Here, we present a set of measurements for four surface-pressure amplitudes and provide a detailed quantitative examination of the various behaviours. At low speeds, the pattern resembles the stationary state (U = 0), essentially a circular dimple located directly under the pressure source (called a state I response). At a critical speed, but still below c(min), there is an abrupt transition to a wave-like state (state II) that features a marked increase in the response amplitude and the formation of a localized solitary depression downstream of the pressure source. This solitary depression is steady, elongated in the cross-stream relative to the streamwise direction, and resembles freely propagating gravity-capillary 'lump' solutions of potential flow theory on deep water. Detailed measurements of the shape of this depression are presented and compared with computed lump profiles from the literature. The amplitude of the solitary depression decreases with increasing U (another known feature of lumps) and is independent of the surface pressure magnitude. The speed at which the transition from states I to II occurs decreases with increasing surface pressure. For speeds very close to the transition point, time-dependent oscillations are observed and their dependence on speed and pressure magnitude are reported. As the speed approaches c(min), a second transition is observed. Here, the steady solitary depression gives way to an unsteady state (state III), characterized by periodic shedding of lump-like disturbances from the tails of a V-shaped pattern.